Chaper 9 Ineres Rae Producs Copyrigh c 2008 20 Hyeong In Choi, All righs reserved. 9. Change of Numeraire and he Invariance of Risk Neural Valuaion The financial heory we have developed so far depends heavily on he privileged role he riskless bond price B plays. Namely, suppose S is a radable asse price, say, sock price. Then he maringale measure Q is defined o be he measure ha makes he discouned sock price processes S = S a maringale. In he parlance of finance, B plays B he role of he so-called numeraire, which, loosely speaking, provides a means of equaing wo differen moneary values measured a wo differen imes by aking ino accoun of he progression of ime. The special privilege of B may be jusified, however feebly, if he shor rae is a consan or even a deerminisic funcion of. However, when erm srucure models which are by naure sochasic are used, B is no, in essence, any differen from oher bond prices or even oher radable asses. For example, we may as well use C = p(, T ) as a numeraire for T. In his secion, we in fac do p(0, T ) jus ha, and see wha is changed in finance. So suppose now ha S is a price of some radable securiy; le B be he price of riskless bond; and le C be anoher radable asse price such ha C > 0 almos surely. For he purpose of normalizaion, we furher assume ha C 0 =. Then we have hree asses; B, C, and S. We also assume ha here is no arbirage in his marke so ha a maringale measure Q ha makes, C, and S simulaneously Q-maringale B B
9.. CHANGE OF NUMERAIRE AND THE INVARIANCE OF RISK NEUTRAL VALUATION 247 exiss. Le X be any European coningen claim wih expiry T. Thus X F T, and by he Risk Neural Valuaion Principle is value V a ime is given by X V = B E Q F. B T On he oher hand C may very well be used as a numeraire. Assuming so, le us now consruc a new measure Q wih respec o which B,, and S are maringales. Since B is numeraire wih C C corresponding maringale measure Q, C is a Q-maringale. For any B T, le us define he measure Q on F by d Q = C B dq. Then { Q } is a family of measures on {F } ha are consisen in he following sense: Le s <, and le A F s, hen obviously A F also. Now Q s (A) = = = A A A = Q (A). d Q s = C E Q F s B C dq B C s dq ( by definiion of A B Q s ) s dq ( C B is a Q-maringale) ( by definiion of condiional expecaion) Thus Q s and Q coincide on F s. This way his family of measures can be pieced ogeher o define a measure Q = Q T on F T so ha Q ( T F = Q. One can in fac exend his family o Q on F = σ F ), bu for our purpose Q s enough. Noe also ha we 0 have already seen his way of piecing ogeher a consisen family of measures. Le we now see ha S C is a Q-maringale. Le s <, and
9.. CHANGE OF NUMERAIRE AND THE INVARIANCE OF RISK NEUTRAL VALUATION 248 A F s, Then S s d A C Q = s = = = = = S s C s dq ( A C s B Q Fs = Q s ) s A S s B s dq S E Q F s B A S B dq A S A A B d B C Q S d C Q. dq ( S B is a Q-maringale) ( by definiion of condiional expecaion) ( On F F s, d Q = C B dq) As S s F s, we have by he definiion of condiional expecaion, C s S E Q F s = S s. C C s Similarly, one can easily see ha B C is a Q-maringale. Wih his, we are ready o prove he following imporan resul. Proposiion 9.. Le S, B, C, Q and Q be as defined above. Suppose X F T. Define V = B E Q X B T F Then V = Ṽ. Ṽ = C E Q X F. C T Proof. Le A F. Then C X E A B Q F dq C T X = E Q F d A C Q ( Q F = Q and d Q = C dq) T B X = d A C Q ( by definiion of condiional expecaion) T X C T = dq ( On F T F, d A C T B Q = C T dq) T B T X = dq. B T A
9.2. FORWARD PRICE AND MEASURE 249 Since C X E B Q F F and he above equaliy holds for all C T A F, we have, by definiion of condiional expecaion, X E Q F = C X E B T B Q F. C T The above proposiion was saed and proved for a paricular measure Q. Bu here may be several maringale measures for he numeraire C in case he marke is no complee. However, even in his case, we have shown in he previous chapers ha he risk neural valuaion does no depend on any paricular maringale measure chosen. Therefore we have he following imporan Theorem 9.2. (Invariance of Risk-Neural Valuaion under Numeraire Change) Le B and C be radable securiies ha are posiive almos surely. Furhermore assume ha B 0 = C 0 =. Le Q (resp. Q) be a measure ha makes all radable securiies discouned by B (resp. C ) Q(resp. Q)-maringale. Le X FT be any European coningen claim. Then X X B E Q F = C E B Q F. T C T 9.2 Forward Price and Measure Previously, we have discussed abou he forward prices. In his secion, we look a he forward conrac involving general European coningen claim. Le X be a European coningen claim wih expiry T 0. A forward conrac a ime is an agreemen enered a o buy or sell he coningen claim a ime T < T 0 a a price deermined a ime. 0 T T 0 presen agreemen enered; price is fixed acual delivery and paymen occur expiry X The quesion is wha should he fair price be a ime, and he obvious answer o i should be he price K given by he formula below. E Q X K B T F = 0, (9.)
9.2. FORWARD PRICE AND MEASURE 250 where B is he usual riskless bond price, and Q is he maringale measure corresponding o he numeraire B. Solving (9.) for K and muliplying B on he numeraor and he denominaor, we have B E X Q B T F K = B E Q B T F = V p(, T ). (9.2) X The las equaliy is obained by considering V = B E Q F B T is he value a of he coningen claim X and B E Q F is B T he value a of he coningen claim ha pays a ime T, which is exacly p(, T ). We call K he forward price a of he conrac enered a for he delivery of X a T and denoe i by F = F (X; T ) Suppose now we choose C = p(, T ) he normalized bond price p(0, T ) as an anoher numeraire, and we le P T be he corresponding maringale measure. This maringale measure P s called a forward measure. Then we have Theorem 9.3. F = E PT X F. V Proof. We showed ha F =. By he invariance of risk neural p(, T ) valuaion under numeraire change, we have X V = C E PT F. C T Bu C T = p(t, T ) p(0, T ) = p(0, T ), which is a known consan, and C = p(, T ). Therefore we have p(0, T ) V = p(, T )E PT X F. This heorem says ha he forward price F is he expecaion a ime of X under he forward measure F T. The argumen in he proof is also resaed as he following: If he binding agreemen on his conrac is made a an earlier ime s <, is value F s a s can be calculaed by F s = E PT X F s = E PT F F s.
9.2. FORWARD PRICE AND MEASURE 25 Theorem 9.4. (Valuaion of Coningen Claim wih he Forward measure) Le X F T, hen is value V a is given by V = p(, T )E PT X F. The meri of his heorem is ha in order o value coningen claims i is NOT necessary o know B. This gives remendous advanage because finding B using he general form of erm srucure model is by no means easy. This fac also comes in handy in he Mone Carlo simulaion, he subjec which we will no ouch in deail here. Le us now examine how he forward rae behaves under he forward measure P T. We have seen ha under he usual maringale measure Q wih numeraire B he bond price saisfies he following sochasic differenial equaion dp(, T ) = p(, T )r d + S(, T )dw, for some S(, T ), regardless of which erm srucure model was used. Le us also noe ha he forward measure P s defined by piecing ogeher measures P given by where dp = ζ dq, ζ = p(, T ) p(0, T )B. Eiher using he fac ha ζ is a Q-maringale or by direc compuaion, i is easy o see ha dζ = ζ S(, T )dw = ζ ( S(, T ))dw ζ 0 =. Thus ζ is an exponenial maringale ha is used in he Girsanov machinery, which says ha he Brownian moion W corresponding o P is defined by d W = dw S(, T )d. (9.3) Now i is easy o check ha d log p(, T ) = r 2 S2 (, T ) d + S(, T )dw.
9.3. GENERALIZED BLACK-SCHOLES FORMULA 252 Taking T and inerchanging he order of differenial we have df(, T ) = d log p(, T ) T = d log p(, T ) T = S(, T ) S S (, T )d T T (, T )dw = S T (, T ) dw S(, T )d = σ(, T )d W, (9.4) where he las equaliy is due o (9.3) and he fac ha σ(, T ) = S (, T ). T This sochasic differenial equaion (9.4) for f(, T ) means ha f(, T ) is a maringale under P T measure. Thus we have Theorem 9.5. (i) For each fixed T, f(,t) is a P T -maringale, i.e., f(s, T ) = E PT f(, T ) F s, (ii) f(, T ) = E PT r T F. The proof is obvious, bu he inerpreaion of his heorem has some ineresing implicaion, namely, he forward rae f(, T ) is he expecaion of he shor rae seen a ime. 9.3 Generalized Black-Scholes Formula A financial asse ha can be bough and sold immediaely in he marke in exchange for cash paymen is called a spo ase. Typical of such asses are socks and bonds 2. Le S be he price process of a spo asse. If ha asse is a sock, S is he usual price process; if i is a T -bond 3, S here sands for p(, T ). As usual, we assume ha S is a semi-maringale wrien as ds = S (r d + β dw ). (9.5) 2 I is cusomary o rea bonds as belonging he a separae asse class, i.e., he fixed-income asses. Bu for our purpose we rea bonds as spo asses. 3 T-bond is a bond wih mauriy T.
9.3. GENERALIZED BLACK-SCHOLES FORMULA 253 We furhermore assume σ is a deerminisic funcion of alhough i may involve some deerminisic parameer like T in he case of T - bond. Here W is assumed o be he Brownian moion wih respec o he maringale measure. Le X be a European opion on such asse wih expiry T. Then we have, as usual, X = (S T K) +. In his secion, we show how o derive pricing formula by uilizing he machinery developed so far in his chaper. Le F = F (, T ) be he forward price of he forward conrac wih he delivery a ime T. Then by (9.2), we have F = Noe ha in he case of T -bond (9.6) is S p(, T ). (9.6) F = p(, T ) p(, T ) (9.7) By Theorem 9.3, F is a P T -maringale, where P s he forward measure corresponding o he delivery ime T. This fac can be argued direcly as follows: firs, S is he price process of a radable asse; second, (9.6) says ha F is S discouned by p(, T ); hird, P s he maringale measure corresponding o he numeraire p(, T )/p(0, T ); herefore, F mus be a maringale wih respec o P T. Therefore by he Girsanov machinery (9.5) is ransformed ino df = F σ dw T (9.8) where W s a P T -Brownian moion. As usual, le p(, T ) saisfy he following SDE dp(, T ) = p(, T )r d + S(, T )dw where W is he Brownian moion wih respec o he maringale measure. Then upon aking he logarihm of S /p(, T ), hen aking d and collecing he coefficien of dw and making use of (9.3), we can conclude ha σ = β S(, T ). (Be careful o disinguish his σ from he volailiy σ(, T ) of he HJM model.) Upon inegraing (9.8), we have F = F 0 exp 2 0 σ 2 udu + 0 σ u dw T u (9.9)
9.3. GENERALIZED BLACK-SCHOLES FORMULA 254 Le Q be he usual maringale measure for S. Then by our riskneural valuaion principle, he value a = 0 of X is given by V 0 = E Q (ST K) + /B T = E Q (ST K)/B T {ST K} = E Q ST /B T {ST K} KEQ /BT {ST K} Noe ha = I I 2. I = {S T K} S T /B T dq. If one uses S /S 0 as a numeraire 4, he corresponding measure Q = Q T becomes d Q = d Q T = S T dq. S 0 B T Therefore I = S 0 d Q(S T K). On he oher hand, he forward measure P on F is defined using p(, T )/p(0, T ) as he numeraire, so dp = p(, T ) p(0, T )B dq. In paricular Therefore Thus we have dp T = I 2 = K {S T K} = Kp(0, T ) p(0, T )B T dq. B T dq {S K} = Kp(0, T )P T (S T K). p(0, T )B T dq V 0 = S 0 Q(ST K) Kp(0, T )P T (S T K) (9.0) Now noe ha F T = S T. Thus S T K if and only if F T K, which is equivalen o saying ha F 0 exp T T σ 2 d + σ dw T K. 2 0 0 Taking logarihm and rearranging erms, i is equivalen o T 0 σ dw T log F 0 K + 2 4 We need o assume S > 0 almos surely for all. T 0 σ 2 d (9.)
9.3. GENERALIZED BLACK-SCHOLES FORMULA 255 Lemma 9.6. For deerminisic σ, T σ dw T is a Gaussian random variable wih mean 0 and variance T 0 σ2 d. Proof. Wrie T 0 σ dw T i σ i W T i, where 0 = 0 < < < n = s a pariion of 0, T. Now each W T i has mean 0 and is independen of W T j for i j. Therefore is variance is E PT i σ i W T i 2 = i σ 2 i i. The proof follows by passing o he limi as we ake finer and finer pariions. Rewrie (9.) by dividing by 0 σ2 d, we have Z = T 0 σ dw T T 0 σ2 d log F 0 K + 2 T o σ2 d 0 σ2 d. Since he lef hand side of he above formula is a sandard N(0, ) random variable Z, we have where Therefore which implies ha P T (S T K) = P T (F T K) = d 2 2π e 2 /2 d, d 2 = log F 0 K T 2 0 σ2 d. 0 σ2 d P T (S T K) = N(d 2 ), I 2 = Kp(0, T )N(d 2 ) (9.2) Le us now look a I. Noe ha ( Q(S T K) = Q P (T, T ) ) S T K ( = Q G T ), K
9.3. GENERALIZED BLACK-SCHOLES FORMULA 256 where G = p(,t ) S = F. Recall ha Q is he measure goen by using S /S 0 as he numeraire. Thus G mus be Q-maringale as i is p(, T ) discouned 5 by S. Now check ha dg = df = F σ dw + (somehing)d. On he oher hand, since G is a Q-maringale, he drif erm mus disappear when dg is wrien in erms of d W, where W is he Q-Brownian moion. Thus dg = G σ d W (9.3) Upon inegraing (9.3), we have G T = G 0 exp T T σ 2 d σ d W 2 0 0 Thus i is easy o check ha G T /K if and only if T ( σ d W log 0 G 0 K ) + T σ 2 d. 2 0 (9.4) T Again dividing by 0 σ2 d and noing ha Z = T 0 σd W T / is a N(0, ) random variable, we can conclude ha where Therefore Q(F T K) = Q(Z d ) d = log ( ) F 0 K + 2 = N(d ) T 0 σ2 d T 0 σ2 d. Hence from (9.2) and (9.5), V 0 becomes 0 σ2 d I = S 0 N(d ) (9.5) V 0 = S 0 N(d ) Kp(0, T )N(d 2 ). Proposiion 9.7. The value a = 0 of he European call opion X = (S T K) + is given by V 0 = S 0 N(d ) Kp(0, T )N(d 2 ), 5 Discouning has o be done by S /S 0. Bu since S 0 is a consan, he asserion is sill valid.
9.3. GENERALIZED BLACK-SCHOLES FORMULA 257 where d = d 2 = ( ) log S0 Kp(0,T ) + T 2 0 σ2 d, T 0 σ2 ( ) d log S0 Kp(0,T ) T 2 0 σ2 d. T 0 σ2 d If we do everyhing from ime insead of 0, we have he following. Theorem 9.8. The value C a of he European call opion X = (S T K) + is given by C = S N(d ) Kp(, T )N(d 2 ) where d = d 2 = ( ) log S Kp(,T ) + T 2 σudu 2, T σudu 2 ( ) log S0 2 σudu 2 Kp(,T ) T σ 2 udu T. To ge he formula for he pu opion, one can uilize he pu-call pariy. Le C and P be he values a of he call and pu opions, respecively. Then i is easy o check ha C T P T = S T K. Thus by he usual risk neural valuaion, B E Q CT /B T F B E Q PT /B T F = B E Q ST /B T F KB E Q /BT F, where Q is he usual maringale measure. Therefore, C P = S Kp(, T ). From his we have he following resul. Theorem 9.9. The value P a of he European pu opion Y = (K S T ) + is given by P = S N( d ) + Kp(, T )N( d 2 ).
9.4. POTPOURRI OF INTEREST-RATE PRODUCTS 258 9.4 Popourri of Ineres-rae Producs For he res of his chaper, we briefly go over some of he more imporan financial produc relaed o he ineres rae, or he erm srucure of ineres rae in general. In paricular when we say p(, T ), f(, T ) and he likes, we always assume, hough acily, he underlying erm srucure model. As a general rule, all of hem are reducible o he bond prices or he opions hereof. For ha reason, we express hem in erms of he bond prices, which can be calculaed, a leas in principle, via he erm srucure model. 9.4. Par bond wih coupons The bond wih coupons is an insrumen (conac) ha pays fixed amoun of cash as principal and he ineres a mauriy. 0 = T 0 T T 2 T N Fix a ime = T 0 = 0. Suppose he ime a which he bond pays ineres is (i=,..., N), and le T N be he ime of mauriy and T 0 = 0 is he presen. A T N, he bond pays he ineres and he principal which is normalized o be. Suppose k is he nominal (coupon) ineres rae of he bond, and le δ = n, where n is he number of imes he bond pays he ineres per year. (We always scale he ime so ha one year is se o be.) The ineres paymen is kδ. For example, suppose he nominal ineres rae is 0% and i pays he ineres quarerly (four imes a year), hen he ineres paymen is kδ = 0. 0.25 = 0.025. Since he bond also pays he principal a T N, he cash sream is as depiced below. kδ kδ kδ kδ + 0 = T 0 T T 2 T N The easies way of valuing his bond is o rea each paymen a ime as separae zero coupon bond. Thus he value of he paymen
9.4. POTPOURRI OF INTEREST-RATE PRODUCTS 259 a ime (i < N) has value kδp(, ) a ime. Thus he oal value of his bond a ime 0 is N p(0, T N ) + kδ p(0, ). (9.6) i= The par bond is he bond whose value a ime of issuance, i.e., = 0 is equal o he principal paymen a mauriy. Thus in order o do so, he issuer has o se he nominal ineres rae k in such a way, he oal value of he bond given by (9.6) is. Therefore, he nominal ineres rae k is calculaed as 9.4.2 Floaing-rae bond k = p(0, T N) δ N i= p(0, ). The floaing-rae bond is he bond ha is he same as he fixed-rae bond excep ha he ineres paymen varies according o pre-agreed arrangemen. A ypical manner in which he ineres paymen is deermined is as follows: Le S < T. The ineres paymen a ime s deermined by he marke rae a ime S. Thus suppose one invess A a ime S and ges paid B, hen he annualized nominal ineres rae k during his period is k = B A (T S)A. In paricular, if A = p(s, T ), and B = p(t, T ) =, hen k = p(s, T ) (T S)p(S, T ) = T S p(s, T ). (9.7) This ype of mechanism is generally adoped in he LIBOR 6 deerminaion. Now suppose T,, T N are he imes of ineres paymen and T N is he ime of mauriy a which ime he final ineres and he principal (=) paymens are made. Assume also ha he paymen is done regularly so ha = δ for some consan for all i. For simpliciy le us also assume ha = T 7 0 and T T 0 = δ. 8 Le us denoe he annualized nominal ineres rae beween and by L( ). Then (9.7) can be rewrien as L( ) = k = δ p(, ) (9.8) 6 London Inerbank Offered Rae 7 We do no assume T 0 = 0. 8 I is no necessary o assume he regular paymen, inerval nor o se = T 0. Bu for simpliciy we make such assumpions. Everyhing done here works for irregular inerval wih minor modificaion.
9.4. POTPOURRI OF INTEREST-RATE PRODUCTS 260 for i =,, N. Noe ha he ineres paymen a ime is δl( ) = p(, ). Since i is a coningen claim a, is value a T 0 is given by ( ) / V i = B T0 E Q p(, ) B Ti F T 0. Now since p(, ) = B Ti E Q F Ti and p(, ) F Ti, we have = E Q p(, ) F Ti. Thus E Q F T0 = E Q E Q p(, ) F Ti F T0 = E Q p(, ) F T0 Therefore, V i can be wrien as V i = B T0 E Q F T0 B T0 E Q F T0 = p(t 0, ) p(t 0, ) Hence he value V a T 0 of his floaing rae bond is V = p(t 0, T N ) + = p(t 0, T 0 ) =. N (p(t 0, ) p(t 0, )) i= Here he erm p(t 0, T N ) represens he value a T 0 of he principal paymen a T N. The curious fac ha he value a T 0 of he floaing rae bond equals he nominal (i.e., undiscouned) principal paymen can be beer undersood if one considers he following rading sraegy. Suppose one sars wih one dollar a ime T 0 wih which one buys
9.4. POTPOURRI OF INTEREST-RATE PRODUCTS 26 p(t 0, T ) unis of T -bond whose price a ime T 0 is p(t 0, T ). This invesmen becomes worh p(t 0, T ) dollars a ime T as he value of T -bond a T is, i.e., p(t, T ) =. Ou of his one pays δl(t 0 ) = p(t 0, T ) as he ineres a T. (Noe ha his ineres paymen a ime T is precisely he ineres he floaing rae bond pays a ime T.) I is clear ha he remaining sum a T afer he ineres paymen is hen. Wih i, one can repea he invesmen paern. Namely a each ime he ineres paymen is δl( ) = p(, ) and he sum available o inves is wih which one buys p(, + ) unis of + -bond whose price a ime is p(, + ). Repeaing he same way i is easy o see ha a ime T N one sill has one dollar remaining afer paying he ineres, which will be given up as he principal paymen. This way, his rading sraegy precisely replicaes he behavior of he floaing rae bond. 9.4.3 Ineres-rae swap Swap is a generic financial erm ha sands for all sors of conracs ha exchange one kind financial asse wih anoher. The one we will be discussing in his secion is he so-called ineres-rae swap. I works in principle as follows. Suppose A holds a bond ha pays ineres in a fixed-rae, say k per annum. Suppose he ime of ineres paymen is for i =,, N and T N is he ime of mauriy a which i also pays as he principal. For he sake of simpliciy, le us assume T 0 =. Is cash sream consising of ineres only excluding he principal is as in Figure 9.. kδ kδ kδ kδ = T 0 T T 2 T N Figure 9.: Cash sream of ineres rae only. Here δ = + for i =,, N.
9.4. POTPOURRI OF INTEREST-RATE PRODUCTS 262 The value a T 0 of income kδ paid a is clearly kδp(t 0, ). Therefore he oal value a T 0 of such cash sream is kδ N p(t 0, ). (9.9) i= Suppose now B holds a bond ha pays he ineres in floaing rae as described in Subsecion 9.4.2. Then is cash sream minus he principal paymen of a T N is as in Figure 9.2. Since he oal δl(t 0 ) δl(t ) δl( ) δl(t N ) T 0 T T 2 T N Figure 9.2: Cash sream of floaing rae. value a T 0 of he floaing-rae bond a ime T 0 is and he value a T 0 of he principal paid a T N is p(t 0, T N ), he value a T 0 of he ineres only cash sream mus be p(t 0, T N ). (9.20) The (ineres-rae) swap is a conrac ha exchanges hese wo cash sreams. In he parlance of finance he payer of he swap is he one who pays ou fixed-rae cash sream and receives he floaingrae cash sream. The posiion held by he payer is called he payer swap. Similarly, he receiver is he one who receives fixed-rae cash sream in lieu of he floa-rae one, and his posiion is called he receiver swap. I is easy o noe ha he payer/receiver designaion is based on he fixed-rae cash sream. When he swap conrac is enered a T 0, he values a T 0 of boh sreams mus coincide. Therefore kδ Solving for k, we have N p(t 0, ) = p(t 0, T N ). i= which is called he swap rae. k = p(t 0, T N ) δ N i= p(t 0, ),
9.4. POTPOURRI OF INTEREST-RATE PRODUCTS 263 In pracice, a each paymen ime boh paries sele only he difference beween he fixed-rae income and he floaing-rae income wihou exchanging he whole amoun. This pracice is called he neing. This swap conrac exchanges ineres incomes only. Bu in order o calculae he acual amoun paid as ineres one has o calculae i as if i is based on some principal. This figure represening such ficiious principal is called he noional amoun. When he organizaions like BIS 9 repors he amoun of swap conracs ousanding worldwide, i repors his noional amoun. As of December, 200, BIS repors he oal ineres-rae swap ousanding worldwide is 364 rillion dollars, which is really he noional amoun. We will laer come back o he relevance of noional amoun. 9.4.4 Forward (ineres-rae) swap Suppose here is a swap conrac enered a ime T 0 as in he previous subsecion. The conrac of he receiver, called he receiver swap, has is value a ime T 0 equal o X = kδ N p(t 0, ) p(t 0, T N ) i= = P (T 0, T N ) + kδ N p(t 0, ). Suppose he boh paries agree a ime < T 0 o ener ino he swap conrac a T 0 wih he rae k prese a ime. This agreemen is i= agreemen ener ino swap T 0 T T N Figure 9.3: Time of agreemen. called he forward swap. This X F T0 is a coningen claim wih expiry T 0. Then applying he lemma below, is value V a is given 9 Bank for Inernaional Selemens.
9.4. POTPOURRI OF INTEREST-RATE PRODUCTS 264 by V = B E Q X/B T0 F = B E Q p(t 0, T N )/B T0 F +kδb N i= B E Q /B T0 F = p(, T N ) + kδ Lemma 9.0. For T < T 2, E Q p(t 0, )/B T0 F N p(, ) p(, T 0 ) (9.2) i= B E Q p(t, T 2 )/B T F = p(, T2 ). Proof. Inuiively, his is clear, since his is he value a of he coningen claim ha pays T 2 -bond a T, which simply mus be he T 2 -bond even a. Thus is value a mus be p(, T 2 ). To prove i analyically, recall ha ( T ) B /B T = exp r s ds and P (T, T 2 ) = B T E Q /B T2 F T Thus B E Q p(t, T 2 )/B T F = E Q exp = E Q B /B T p(t, T 2 ) F = E Q exp ( T ( = E Q E Q exp ( T2 T ) ( T2 r s ds E Q exp T2 = E Q exp ( T2 r s ds ) F = p(, T 2 ). ) FT r s ds. T FT F r s ds) ) FT F r s ds ( T ) r s ds F T
9.4. POTPOURRI OF INTEREST-RATE PRODUCTS 265 As usual, he value V (as in (9.2)) of his forward swap conrac mus be zero a ime. Therefore, upon seing V = 0 in (9.2) and solving for k, we ge k = p(, T 0) p(, T N ) δ N i= p(, T, i) which is he rae prese a wih which he swap conrac is o be enered ino a ime T 0. k is called he forward swap rae. Dividing he denominaor and he numeraor by p(, T 0 ), we have k = F (T 0, T N ) δ N i= F (T 0, ), where F (T 0, ) = p(, )/p(, T 0 ) is he forward price a for he purchase of T 0 -bond in reurn for a. 9.4.5 Bond opion Le p(, T ) be he price a of he bond wih mauriy T. In his secion, we sudy how o value he opions on i. In paricular, le X be he call opion on i wih srike price K a he expiry T T. Then X F T and is given by X = ( p(t, T ) K ) +. We apply he generalized Black-Scholes formula in Secion 9.3 o derive a valuaion formula for X. Le F = F (, T ) be he price a of he forward conrac ha delivers T -bond a T T. Then he forward price a is given by F = F (, T ) = p(, T ) p(, T ). Assume, as we have done in Chaper 8, ha p(, U) saisfies he following dynamics dp(, U) = p(, U) m(, U)d + S(, U)dW for any mauriy dae U, where W is he Brownian moion wih respec o a physical measure. We furhermore assume S(, U) is a deerminisic funcion of wo variables and U. Then i is easy o check by brue force ha ( ) p(, T ) d p(, T ) = p(, T ) p(, T ) (S(, T ) S(, T ) ) dw + ( S(, T ) 2 S(, T )S(, T ) ) d,
9.4. POTPOURRI OF INTEREST-RATE PRODUCTS 266 where T S(, T ) = σ(, u)du, (9.22) as in Secion 8.6. By he resul of Secion 9.2, he Brownian moion W T corresponding o he forward measure P T saisfies dw T = dw S(, T )d. Therefore we have ( ) p(, T ) d p(, T ) In oher words = p(, T ) S(, T ) S(, T ) dw T. p(, T ) df = F S(, T ) S(, T ) dw T. (9.23) This forward price dynamics can be also seen as follows. Firs, F is a maringale wih respec o P T. Therefore F mus be of he form df = F σ dw T, for some σ. We hen have o check wha his σ has o be. For ha one akes he logarihm of F = p(, T )/p(, T ), akes d, and collecs volailiy erms, i.e., he coefficiens of dw T, o see ha σ = S(, T ) S(, T ) = T T σ(, u)du, where he las equaliy is due o (9.22). Invoking Theorem 9.8, we have he following. Theorem 9.. Le X be he call opion on he T -bond. Assume ha he srike price of X is K and he expiry T T. Assume furher ha for any U he bond price p(, U) saisfies he SDE dp(, U) = p(, U) m(, U)d + S(, U)dW, where S(, U) is a deerminisic funcion of and U, where W is he Brownian moion wih respec o a physical measure. Then he value C a of he call opion is given by C = p(, T )N(d ) Kp(, T )N(d 2 ),
9.4. POTPOURRI OF INTEREST-RATE PRODUCTS 267 where d = d 2 = ( log p(,t ) Kp(,T ) ( log p(,t ) Kp(,T ) ) + T 2 T σudu 2 ) 2 T σ 2 udu T σudu 2, σudu 2, and T σ = S(, T ) S(, T ) = σ(, u)du. T By invoking Theorem 9.9, he pu opion formula can also be obained. Theorem 9.2. Le X be he pu opion on he T -bond. Assume ha he srike price of X is K and he expiry T T. Assume furher ha for any U he bond price p(, U) saisfies he SDE dp(, U) = p(, U) m(, U)d + S(, U)dW where S(, U) is a deerminisic funcion of and U,where W is he Brownian moion wih respec o a physical measure. Then he value P a of he pu opion is given by P = p(, T )N( d ) + Kp(, T )N( d 2 ), where d = d 2 = ( log p(,t ) Kp(,T ) ( log p(,t ) Kp(,T ) ) + T 2 T σudu 2 ) 2 T σ 2 udu T σudu 2, σudu 2, and T σ = S(, T ) S(, T ) = σ(, u)du. T 9.4.6 Opion on coupon bond In his secion, we sudy opions on coupon bonds. Unlike he zerocoupon bond we have sudied in he previous secion, he presence of he sream of cash paymens complicaes he problem. Le, for i =, n, be he ime of ineres paymen and T n he mauriy of
9.4. POTPOURRI OF INTEREST-RATE PRODUCTS 268 kδ kδ kδ kδ + T T a T T a T n Figure 9.4: Timeline and cash flows. his bond. Le T be he expiry of he opion, which falls in he ime inerval T a, T a ), i.e., T a T < T a. Le X be he call opion on his bond wih he srike price K a he expiry T. Then as he owner of X is eniled, a T, o he whole of subsequen cash flows, X as an F T -random variable mus be of he form ( + n X = p(t, T n ) + kδ p(t, ) K). As K is relaed o his se of cash flows, he resul of he previous subsecion canno be direcly applied, which makes i a very difficul problem o value X in a general seing. However in a cerain case i can be done wihou oo much complicaion, say, if he bond price p(, T ) can be given as a funcion p(, T, r) of hree deerminisic variables, T, r, where r sands for he shor rae. Noe ha p(, T, ) = 0, which conforms o he inuiion ha he bond should have no value if he ineres rae (hence he discoun rae) is infinie, i.e., he fuure cash flows should amoun o zero presen value. Define i=a V (T, r) = p(t, T n, r) + kδ n p(t,, r). i=a Clearly V (T, 0) = + kδ(n a + ). Since no one in he righ frame of mind will pay V (T, 0) or more a T for he remaining subsequen cash flow of his bond, he srike price K should have been se as smaller han V (T, 0). Hence i is very reasonable o assume ha V (T, ) = 0 < K < V (T, 0). Therefore, by coninuiy, here mus be r such ha V (T, r ) = K. Le K i be such ha K i = p(t,, r ).
9.4. POTPOURRI OF INTEREST-RATE PRODUCTS 269 Then n V (T, r ) = p(t, T n, r ) + kδ p(t,, r ) n = K n + kδ K i. i=a i=a Furhermore, using he fac ha p(, T, r) and hence V (T, r) is a monoone decreasing funcion of r, i is no hard o check ha for any r, V (T, r) K if and only if p(t,, r) K i for every i = a,, n. Therefore we can wrie X = ( ) + n ( ) +. p(t, T n ) K n + kδ p(t, Ti ) K i Once ( X is broken ) down his way, i can be valued by valuing each + p(t, Ti ) K i separaely, say, using he mehod inroduced in he previous subsecion. The same argumen applies o he pu opion on coupon bond. 9.4.7 Cap and floor 9.4.7. Cap Floaing-rae ineres paymen is a common form of fixed-income invesmen. The problem wih his kind of arrangemen is ha he payer of floaing-rae ineres may be exposed o unexpecedly large ineres paymen. To hedge agains such coningency, one may ener ino a conrac ha limis (caps) he maximum of ineres rae he payer is liable for. Suppose he cash sream is as in Figure 9.5. Suppose he ineres-rae cap is k. According o his arrangemen, if i=a δl(t 0 ) δl(t ) δl( ) δl(t N ) = T 0 T T 2 T N Figure 9.5: Timeline and cash sream. he ineres rae L( ) a is less han k, he payer pays δl( ) as usual. Bu according o his conrac, if L( ) > k, hen he payer is obligaed o pay only kδ, which amouns o he payer s
9.4. POTPOURRI OF INTEREST-RATE PRODUCTS 270 paying δl( ) bu receiving he difference δl( ) kδ from he receiver. In oher words, he cap conrac eniles payer o receive X = δ ( L( ) k ) + (9.24) a each ime. This individual componen o be execued a is called he caple, and he enirey is called he cap. In his subsecion we sudy how o value i. Le p i = p(, ). Then using (9.8) we have X = δ δ = ( ) + k p i ( p i ( + kδ) ) + = ( + kδ)p i (K p i ) +, where K = ( + kδ). Therefore he value V a of his caple X is V = B E Q X F Now = ( + kδ)b E Q p i (K p i ) + F. (9.25) E Q p i (K p i ) + F = E Q E Q B p i (K p i ) + F Ti F = E Q p i (K p i ) + E Q B FTi F = E Q p i (K p i ) + B Ti = E Q (K p i ) + F. ( p i F Ti ) E Q B F Ti F ( B Ti E Q FTi = p i ) Therefore Noe ha V = ( + kδ)b E Q ( K p(ti, ) ) + F. B E Q ( K p(ti, ) ) + F is he value of he pu opion an -bond wih he srike price K a he expiry, which can be evaluaed by he mehod, say, in Subsecion 9.4.5.
9.4. POTPOURRI OF INTEREST-RATE PRODUCTS 27 9.4.7.2 Floor If a cap is like a call opion, a floor is a conrac ha works like a pu opion. so he floor conrac holder receives a Y = δ ( k L( ) ) +. As before, each individual conrac execued a each ime is called he floorle and he enirey of such floorles is called he floor. Alhough i is enirely possible o evaluae Y mimicking wha was done for caple, i is more illusraive o look a he so-called floor-cap pariy. Le Caple() be he value of he caple a and Floorle() ha of he floorle a. Then i is rivial o see ha Floorle( ) Caple( ) = δ ( k L( ) ) = ( + kδ) p i. Thus, upon apply he risk neural valuaion principle, we have Now Floorle() Caple() = ( + kδ)b E Q F B E Q E Q Therefore p i p i F = E Q E Q p i F Ti F = E Q p i E Q B F Ti F = E Q p i = E Q F T I F. ( p i F Ti ) B Ti E Q B F Ti F ( B Ti E Q B F Ti = pi ) Floorle() Caple() = ( + kδ)p(, ) p(, ), (9.26) from which and from he formula for Caple(), Floorle() can be found. Formula (9.26) is called he floor-cap pariy. 9.4.8 Swapion and ec. Swapion is an opion o ener ino a swap conrac a a fuure daa a fixed rae 0 k. Figure 9.6 shows he imeline. The swap conrac 0 The ineres rae k is no he swap rae as described in 9.4.3. I is simply a fixed number agreed upon when he swapion conrac is enered ino.
9.4. POTPOURRI OF INTEREST-RATE PRODUCTS 272 T 0 T T N Figure 9.6: Timeline. is o be enered a T 0 wih he fixed rae k, and he cash flow occurs a imes, i =, 2,, n. The value a T 0 of he receiver swap is and he payoff a T 0 is ( p(t 0, T n ) + kδ p(t 0, T n ) + kδ n p(t 0, ), i= + n p(t 0, ) ). i= This is exacly a call opion on he coupon bond wih coupon rae k whose srike price is and expiry T 0. The valuaion of his conrac can be done using he mehod of 9.4.6. Remark 9.3. Opions on caps and floors are also very popular conracs. An opion on a cap is called a capion, and similarly an opion on a floor is called a floorion. However, he valuaion of such insrumens is more complicaed and is beyond he scope of his lecure. The readers are referred o many advanced exbooks.
EXERCISES 273 Exercises 9.. Le W be a Brownian moion and for each le X be a random variable given by X = 0 e 2u+3 dw u (a) Find he mean and variance of X. (b) For fixed, is X a Gaussian random variable? Jusify your answer. (c) Regarding X as a sochasic process, is X a maringale? If so, why; if no, why no? 9.2. Suppose a floaing rae bond X pays ou ineres δl( ) a ime for i =,..., n such ha δl( ) = p(, ) where p(, T ) is he price a of he zero-coupon bond paying a T. Assume also his bond X pays ou he principal a ime T n. Wha should is value a T 0 be? Jusify your answer. 9.3. Answer he following quesions. (a) Le p(, T ) be he price a of he zero-coupon bond paying a T. Le T < T. Le F (, T ) be he forward price a of he forward conrac delivering he T -bond (i.e., he zero-coupon bond ha pays a T ) a T. Wrie down F (, T ) in erms of zero-coupon bond prices p(, T ) (for various T ). (b) Explain why F = F (, T ) as a sochasic process has o be a maringale wih respec o P T where P s he T -forward measure. 9.4. Le X be a sochasic process given by dx = a d + b dw, where a and b are consans and W is a Brownian moion. Le Y be a random variable defined by Y = T 0 et dx, where s a fixed posiive number. (a) Prove ha Y is a Gaussian random variable. (b) Calculae he mean and he variance of Y.
EXERCISES 274 9.5. Le < T < T 2, and le S be a sock price process. (a) Wha is he forward price F (T, T 2 ) a T of he conrac o deliver his sock a T 2? (b) Le X F T2 be a coningen claim ha pay F (T, T 2 ) a ime T 2. Wha is he value of X a? 9.6. Le T 0 < T < < T N be given such ha = δ for i = 0,, N. The LIBOR rae L( ) beween and is given by L( ) = δ p(, ), where p(,t) is he value a of he zero-coupon bond paying a T. (a) The caple( ) a wih he cap rae k is defined by caple( ) = δ(l( ) k) +. Describe he floorle conrac foorle( ) wih he floor rae k. (b) Sae and prove he floor-cap pariy formula.